What is the difference of Rotating counterclockwise by origin by $\theta$ and rotating clockwise by origin through $\theta$? And what are their formulas? I know that for counterclockwise rotation, the formula is
$$T = \left(\begin{array}{rr}\cos(\theta) & -\sin(\theta)\\\sin(\theta) & \cos(\theta) \end{array}\right)$$
But what is the formula for clockwise rotation?
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A clockwise rotation by angle $\theta$ is the inverse of a counterclockwise rotation by angle $\theta$ so you need to invert the original matrix $$\boldsymbol T:=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$$ Noting that this is an orthogonal matrix (all matrices associated with rotations are orthogonal), i.e., $\boldsymbol T^\intercal\boldsymbol T=\mathbf I$ (the identity matrix), the inverse is $$\boldsymbol T^{-1}=\boldsymbol T^\intercal=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}.$$
You have a mapping $T:[0,2\pi) \rightarrow \mathbb{R}^2$. If you want to rotate clockwise, then just rotate by $-\theta$ radians. So $$ S(\theta)=\left[ \begin{array}{cc} cos(\theta) & sin(\theta)\\ -sin(\theta) & cos(\theta) \\ \end{array} \right] $$ where I'm using the fact that $cos(.)$ is an even function and $sin(.)$ is an odd function.